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I've recently learned that If you move a conductor through a magnetic field, an emf is induced across the ends of the conductor:

$E = BLv $

I've also been told that the same is true for a coil but that the equation is:

$E = BLvN$

And that for a coil, there is only an emf induced when the current is entering or leaving the magnetic field as once the coil is inside, there is no change in magnetic flux linkage. I've been told that this is not the case with just a wire moving through a magnetic field, but I haven't been told why and I can't seem to figure it out.

I've thought about it, and I think it perhaps has something to do with the area covered by the wire moving at speed $v$ per unit time compared to the area covered by a coil moving at the same speed $v$ in the same unit time.

Unfortunately, I can't quite work it out.

Thank you.

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You're almost right. The induced electromotive force due to external magnetic field is proportional to rate of change of magnetic flux through the circuit. In case of the coil moving in a uniform field, the flux through it (proportional to the number of turns) does not change in time. In case of the wire, the flux changes, since the circuit's area grows or shrinks.

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